#8521623

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,10)(3,8,11,5)(4,6,9,15)(12,13,16,14)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3,7,11)(2,8,10,5)(4,12,9,16)(6,13,15,14)(18,23)(19,26)(25,28)(27,32)(34,35)(40,43)(41,42)(47,48);(3,11)(5,8)(12,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,4)(2,6)(3,12)(5,14)(7,9)(8,13)(10,15)(11,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36,39,46)(34,40,35,43)(37,44,38,45)(41,47,42,48);(1,5)(2,3)(4,14)(6,12)(7,8)(9,13)(10,11)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,6)(2,9)(3,13)(4,10)(5,12)(7,15)(8,16)(11,14)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8428819);